A FREQUENCY SPACE FOR THE HEISENBERG GROUP

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ANNALES DE

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Hajer Bahouri, Jean-Yves Chemin & Raphaël Danchin A frequency space for the Heisenberg group

Tome 69, n^o1 (2019), p. 365-407.

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A FREQUENCY SPACE FOR THE HEISENBERG GROUP

by Hajer BAHOURI,

Jean-Yves CHEMIN & Raphaël DANCHIN (*)

Abstract. — We revisit the Fourier analysis on the Heisenberg groupH^d. Starting from the so-called Schrödinger representation and taking advantage of the projection with respect to the Hermite functions, we look at the Fourier transform of an integrable functionf,as a function^fbHon the setHe^d^def=N^d×N^d×R\{0}. After observing that ^fbH is uniformly continuous onHe^d equipped with an appropriate distance ^d,bwe extend the definition of^fbH to the completionbH^dofHe^d.This new point of view provides a simple and explicit description of the Fourier transform of integrable functions, when the “vertical” frequency parameter tends to 0.As an application, we prepare the ground for computing the Fourier transform of functions onH^dthat are independent of the vertical variable.

Résumé. — On propose une nouvelle approche de la théorie de Fourier sur le groupe de Heisenberg. En utilisant la représentation de Schrödinger et la projection sur les fonctions de Hermite, la transformée de Fourier d’une fonction intégrable est définie comme une fonction sur l’ensemble eH^d^def=N^d×N^d×R\ {0}.Cette fonction étant uniformément continue sur eH^d muni d’une distance adéquate, on peut l’étendre par densité sur le complété bH^d de eH^d. Ce nouveau point de vue fournit une description simple de la limite de la transformée de Fourier des fonctions intégrables lorsque la « fréquence verticale » tend vers 0.On dispose ainsi d’un cadre adéquat pour calculer par exemple la transformée de Fourier d’une fonction indépendante de la variable verticale.

Keywords:Fourier transform, Heisenberg group, frequency space, Hermite functions.

2010Mathematics Subject Classification:43A30, 43A80.

(*) The authors are indebted to Fulvio Ricci for enlightening suggestions that played a decisive role in the construction of this text, and to Nicolas Lerner for enriching discussions. A determining part of this work has been carried out in the exceptional environment of theCentre International de Rencontres Mathématiquesin Luminy. The third author has been partially supported by theInstitut Universitaire de France.

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1. Introduction

Fourier analysis on locally compact Abelian groups is by now classical matter that goes back to the first half of the 20th century (see e.g. [16] for a self-contained presentation).

Consider a locally compact Abelian group (G,+) endowed with a Haar measure µ, and denote by (G,b ·) the dual group of (G,+) that is the set of characters onGequipped with the standard multiplication of functions.

By definition, the Fourier transform of an integrable functionf : G→C is the continuous and bounded functionfb: Gb →C(also denoted by Ff) defined by

(1.1) ∀γ∈G,b fb(γ) =Ff(γ)^def= Z

G

f(x)γ(x)dµ(x).

Being also a locally compact Abelian group, the “frequency space”Gbmay be endowed with a Haar measureµ.b Furthermore, one can normalizeµbso that the followingFourier inversion formulaholds true for, say, all function f in L¹(G) withfbinL¹(G):b

(1.2) ∀x∈G, f(x) =

Z

Gb

f(γ)b γ(x)dµ(γ).b As a consequence, we get the Fourier–Plancherel identity (1.3)

Z

G

|f(x)|²dµ(x) = Z

Gb

|fb(γ)|²dµ(γ)b for all f in L¹(G)∩L²(G).

The Fourier transform on locally compact Abelian groups has a number of noteworthy properties that we do not wish to enumerate here. Let us just recall that it changes convolution products into products of functions, namely

(1.4) ∀f ∈L¹(G), ∀g∈L¹(G), F(f ? g) =Ff· Fg.

In the Euclidean caseG=Rⁿ the dual group may be identified to (Rⁿ)^? through the map ξ 7→ e^{ihξ,· i} (where h ·,· i stands for the duality bracket between (Rⁿ)^? andRⁿ), and the Fourier transform of an integrable functionf may thus be seen as the function on (Rⁿ)^? (usually identified toRⁿ) given by

(1.5) F(f)(ξ) =fb(ξ)^def= Z

Rⁿ

e^−ihξ,xif(x)dx.

Of course, we have (1.4) and, as is well known, if one endows the frequency space (Rⁿ)^? with the measure _(2π)¹ndξ then the inversion and Fou- rier–Plancherel formulae (1.2) and (1.3) hold true. Among the numerous

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additional properties of the Fourier transform onRⁿ,let us just underline that it allows to “diagonalize” the Laplace operator, namely for all smooth compactly supported functions, we have

(1.6) F(∆f)(ξ) =−|ξ|²fb(ξ).

For noncommutative groups, Fourier theory gets wilder, for the dual group is too “small” to keep the definition of the Fourier transform given in (1.1) and still have the inversion formula (1.2). Nevertheless, if the group has “nice” properties (that we are not going to list here) then one can work out a consistent Fourier theory with properties analogous to (1.2), (1.3) and (1.4) (see e.g. [1, 5, 6, 12, 15, 17, 18, 20] and the references therein for the case of nilpotent Lie groups). In that context, the classical definition of the Fourier transform amounts to replacing characters in (1.1) with suitable families of irreducible representations that are valued in Hilbert spaces (see e.g. [6, 9] for a detailed presentation). Consequently, the Fourier transform is no longer a complex valued function but rather a family of bounded operators on suitable Hilbert spaces. It goes without saying that within this approach, the notion of “frequency space” becomes unclear, which makes Fourier theory much more involved than in the Abelian case.

In the present paper, we want to focus on the Heisenberg group which, to some extent, is the simplest noncommutative nilpotent Lie group and comes into play in various areas of mathematics, ranging from complex analysis to geometry or number theory, probability theory, quantum mechanics and partial differential equations (see e.g. [3, 7, 17, 18]). As several equivalent definitions coexist in the literature, let us specify the one that we shall adopt throughout.

Definition 1.1. — Let σ(Y, Y⁰) = hη, y⁰i − hη⁰, yi be the canonical symplectic form onT^?R^d. The Heisenberg groupH^d is the setT^?R^d×R equipped with the product law

w·w⁰^def= Y+Y⁰, s+s⁰+2σ(Y, Y⁰)

= y+y⁰, η+η⁰, s+s⁰+2hη, y⁰i−2hη⁰, yi where w = (Y, s) = (y, η, s) and w⁰ = (Y⁰, s⁰) = (y⁰, η⁰, s⁰) are generic elements ofH^d.

As regards topology and measure theory on the Heisenberg group, we shall look atH^d as the setR^2d+1,after identifying (Y, s) in H^d to (y, η, s) inR^2d+1.With this viewpoint, theHaar measureonH^dis just the Lebesgue measure onR^2d+1. In particular, one can define the following convolution

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product for any two integrable functionsf andg:

(1.7) f ? g(w)^def= Z

H^d

f(w·v⁻¹)g(v)dv= Z

H^d

f(v)g(v⁻¹·w)dv.

Even though convolution on the Heisenberg group is noncommutative, if one defines theLebesgue spacesL^p(H^d) to be justL^p(R^2d+1),then one still gets the classical Young inequalities in that context.

As already explained above, asH^d is noncommutative, in order to have a good Fourier theory, one has to resort to more elaborate irreducible representations than characters. In fact, the group of characters onH^dis isomet- ric to the group of characters onT^?R^d.Roughly, if one defines the Fourier transform according to (1.1) then the information pertaining to the vertical variablesis lost.

There are essentially two (equivalent) approaches, that are based ei- ther on theBargmann representationor on theSchrödinger representation (see [6]). For simplicity, let us just recall the second one which is the family of group homomorphismsw7→U_w^λ (withλ∈R\ {0}) between H^d and the unitary groupU(L²(R^d)) ofL²(R^d),defined for allw= (y, η, s) inH^d and uin L²(R^d) by

U_w^λu(x)^def= e−iλ(s+2hη,x−yi)u(x−2y).

The classical definition of Fourier transform of integrable functions onH^d reads as follows:

Definition 1.2. — The Fourier transformof an integrable function f on H^d is the family (F_H(f)(λ))_λ∈_R_\{0} of bounded operators on L²(R^d) given by

F_H(f)(λ)^def= Z

H^d

f(w)U_w^λdw.

In the present paper, we strive for another definition of the Fourier transform, that is as similar as possible to the one for locally compact groups given in (1.1). In particular, we want the Fourier transform to be a complex valued function defined on some “explicit” frequency space that can be endowed with a structure of a locally compact and complete metric space, and to get formulae similar to (1.2), (1.3), (1.4) together with a di- agonalization of the Laplace operator (for the Heisenberg group of course) analogous to (1.6).

There is a number of motivations for our approach. An important one is that, having an explicit frequency space will allow us to get elementary proofs of the basic results involving the Fourier transform, just by mim- icking the corresponding ones of the Euclidean setting. In particular, we

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expect our setting to open the way to new results for partial differential equations on the Heisenberg group. Furthermore, our definition will enable us to get an explicit (and comprehensible) description of the range of the Schwartz space by the Fourier transform. As a consequence, extending the Fourier transform to the set of tempered distributions will become rather elementary (see more details in our forthcoming paper [2]).

In the present paper, we will give two concrete applications of our approach. First, in Theorem 2.7, we will provide an explicit asymptotic description of the Fourier transform when (what plays the role of) the vertical frequency parameter tends to 0. Our second application is the extension (also explicit) of the Fourier transform to functions depending only on the horizontal variable (this is Theorem 2.8).

2. Results

Before presenting the main results of the paper, let us recall how, with the standard definition of the Fourier transform in H^d, Properties (1.2), (1.3) and (1.4) may be stated (the reader may refer to e.g. [3, 4, 7, 8, 9, 10, 12, 13, 17, 18, 20] for more details).

Theorem 2.1. — Letf be an integrable function. Then we have (2.1) ∀λ∈R\ {0}, kF_H(f)(λ)k_L(L2)6kfk_L1(H^d)

and, for any functionuinL²(R^d), the mapλ7→F_H(f)(λ)(u)is continuous fromR\ {0} to L²(R^d). For any function f in the Schwartz spaceS(H^d) (which is the classical Schwartz space on R^2d+1), we have the inversion formula:

(2.2) ∀w∈H^d, f(w) = 2^d−1 π^d+1

Z

R

tr U_w^λ−1F_Hf(λ)

|λ|^ddλ ,

where tr(A) denotes the trace of the operator A. Moreover, if f belongs toL¹(H^d)∩L²(H^d)thenF_H(f)(λ)is an Hilbert–Schmidt operator for anyλ inR\ {0}, and we have

(2.3) kfk²_L2(H^d)= 2^d−1 π^d+1

Z

R

kF_H(f)(λ)k²_HS|λ|^ddλ wherek · kHS stands for the Hilbert–Schmidt norm.

We also have an analogue of the convolution identity (1.4). Indeed, as the mapw7→U_w^λ is a homomorphism between H^d and U(L²(R^d)), we get for any integrable functionsf andg,

(2.4) F_H(f ? g)(λ) =F_H(f)(λ)◦F_H(g)(λ).

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Let us next recall the definition of the (sub-elliptic) Laplacian on the Heisenberg group, that will play a fundamental role in our approach. Being a real Lie group, the Heisenberg group may be equipped with a linear space ofleft invariant vector fields, that is vector fields commuting with any left translation τw(w⁰)^def=w·w⁰. It is well known that this linear space has dimension 2d+ 1 and is generated by the vector fields

S^def=∂s, Xj

def=∂y_j+ 2ηj∂s and Ξj

def=∂η_j−2yj∂s, 16j6d.

The Laplacian associated to the vector fields (Xj)16j6d and (Ξj)16j6d

reads

(2.5) ∆_H^def=

d

X

j=1

(X_j²+ Ξ²_j).

As in the Euclidean case (see Identity (1.6)), the Fourier transform allows to diagonalize Operator ∆_H, a property that is based on the following relation that holds true for all functions f and u in S(H^d) and S(R^d), respectively (see e.g. [11, 14]):

(2.6) F_H(∆_Hf)(λ) = 4F_H(f)(λ)◦∆^λ_osc

with ∆^λ_oscu(x)^def=

d

X

j=1

∂_j²u(x)−λ²|x|²u(x).

This prompts us to take advantage of the spectral structure of the har- monic oscillator to get an analog of Formula (1.6). To this end, we need to introduce the family of Hermite functions (H_n)_n∈Nd defined by

H_n^def= 1

2^|n|n!

¹₂

CⁿH₀ with Cⁿ^def=

d

Y

j=1

C_jⁿ^j and H₀(x)^def=π⁻^d² e⁻^|x|

2 2

whereCj

def=−∂j+Mj stands for thecreation operatorwith respect to the j-th variable andMj is the multiplication operator defined byMju(x)^def= xju(x).As usual, n!^def=n1!· · ·nd! and |n|^def=n1+· · ·+nd.

It is well known that the family (H_n)_n∈_Ndis an orthonormal basis of the Hilbert spaceL²(R^d).In particular,

(2.7) ∀(n, m)∈N^d×N^d, (Hn|Hm)L²=δn,m, whereδ_n,m= 1 ifn=m,andδ_n,m= 0 ifn6=m.

Besides, we have

(2.8) (−∂_j²+M_j²)Hn = (2nj+ 1)Hn and −∆¹_oscHn= (2|n|+d)Hn.

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ForλinR\ {0},we further introduce the rescaled Hermite function H_n,λ(x)^def=|λ|^d⁴H_n(|λ|¹²x).

It is obvious that (Hn,λ)_n∈Nd is an orthonormal basis ofL²(R^d),that (−∂_j²+λ²M_j²)Hn,λ= (2nj+ 1)|λ|Hn,λ

and thus

(2.9) −∆^λ_oscH_n,λ= (2|n|+d)|λ|H_n,λ.

We are now ready to give “our” definition⁽¹⁾of the Fourier transform onH^d. Definition 2.2. — Let He^d^def=N^d×N^d×R\ {0}. We denote by wb = (n, m, λ) a generic point of He^d. For f in L¹(H^d), we define the mapF_Hf (also denoted byfb_H) to be

F_Hf : (

He^d −→C

wb 7−→ F_H(f)(λ)Hm,λ|Hn,λ

L².

From now on, we shall use only that definition of the Fourier transform, which amounts to considering the “infinite matrix” of F_Hf(λ) in the orthonormal basis ofL²(R^d) given by (H_n,λ)_n∈_N.To emphasize the likeness with the Euclidean case, we shall rewriteF_Hf in terms of the mean value off modulated by some oscillatory functionswhich may be seen as suitable Wigner distribution functions of the family (Hn,λ)_n∈_Nd,λ6=0, and will play the same role as the characters e^{ihξ,· i} inRⁿ.Indeed, by definition, we have

F_Hf(w) =b Z

H^d×R^d

f(w) e^−isλe−2iλhη,x−yiHm,λ(x−2y)Hn,λ(x)dwdx.

Therefore, making an obvious change of variable, we discover that F_Hf(w) =b

Z

H^d

e^isλW(w, Yb )f(Y, s)dYds with (2.10)

W(w, Yb )^def= Z

R^d

e^2iλhη,ziHn,λ(y+z)Hm,λ(−y+z)dz.

(2.11)

At this stage, looking at the action of the Laplace operator on e^isλW(w, Yb ) is illuminating. Indeed, easy computations (carried out in Appendix) give (2.12) (X_j²+ Ξ²_j) e^isλW(w, Yb )

=−4|λ|(2mj+ 1) e^isλW(w, Yb ).

By summation onj∈ {1, . . . , d},we get (2.13) ∆_H e^isλW(w, Yb )

=−4|λ|(2|m|+d) e^isλW(w, Yb ),

(1)This point of view has been already taken in [19] in connection with uncertainty inequalities on the Heisenberg group.

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from which one may deduce that, wheneverf is in S(H^d) (again, refer to the Appendix),

(2.14) F_H(∆_Hf)(w) =b −4|λ|(2|m|+d)fb_H(w).b

Let us underline the similarity with Relation (1.6) pertaining to the Fourier transform inRⁿ.

One of the basic principles of the Fourier transform on Rⁿ is that “reg- ularity implies decay”. It remains true in the Heisenberg framework, as stated in the following lemma.

Lemma 2.3. — For any non negative integerp, there exist an integerNp

and a positive constantCpsuch that for anywb inHe^dand anyf inS(H^d), we have

(2.15) 1 +|λ|(|n|+|m|+d) +|n−m|^p

|fb_H(n, m, λ)|6CpkfkN_p,S, where k · kN,S denotes the classical family of semi-norms of S(R^2d+1), namely

kfk_N,S^def= sup

|α|6N

(1 +|Y|²+s²)^N/2∂_Y,s^α f _L_∞.

As may be easily checked by the reader, in our setting, there are very simple formulae corresponding to (2.2) and (2.3), if the setHe^d is endowed with the measure dwbdefined by:

(2.16)

Z eH^d

θ(w)db wb^def= X

(n,m)∈N^2d

Z

R

θ(n, m, λ)|λ|^ddλ.

Then Theorem 2.1 recasts as follows:

Theorem 2.4. — The following inversion formula holds true for any functionf inS(H^d):

(2.17) f(w) = 2^d−1 π^d+1

Z eH^d

e^isλW(w, Yb )fb_H(w)db w.b Moreover, for any functionf inL¹(H^d)∩L²(H^d),we have (2.18) kfb_Hk²

L²(e^H^d⁾

= π^d+1

2^d−1kfk²_L2(H^d).

In this new setting, the convolution identity (2.4) rewrites as follows for all integrable functionsf andg:

(2.19) F_H(f ? g)(n, m, λ) = (fb_H·bg_H)(n, m, λ) with (fb_H·bg_H)(n, m, λ)^def= X

`∈N^d

fb_H(n, `, λ)bg_H(`, m, λ).

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The reader is referred to the appendix for the proof.

Next, we aim at endowing the setHe^dwith a structure of a metric space.

According to the decay inequality (2.15), it is natural to introduce the following distanced:b

(2.20) d(bw,b wb⁰)^def=

λ(n+m)−λ⁰(n⁰+m⁰) ₁+

(n−m)−(n⁰−m⁰)|1+|λ−λ⁰|, where| · |₁ denotes the`¹norm onR^d.

At first glance, the metric space (He^d,d) seems to be the natural frequencyb space within our approach. However, it fails to be complete, which may be a source of difficulties for further development. We thus propose to work with its completion, that is described in the following proposition.

Proposition 2.5. — The completion of the setHe^dfor the distancedbis the setHb^d defined by Hb^d^def= N^2d×R\ {0}

∪Hb^d0 withHb^d0

def=R^d∓×Z^d and R^d∓

def= ((R−)^d∪(R+)^d).On Hb^d, the extended distance (still denoted byd)b is given by

d((n, m, λ),b (n⁰, m⁰, λ⁰)) =

λ(n+m)−λ⁰(n⁰+m⁰) ₁ +

(m−n)−(m⁰−n⁰)|1+|λ−λ⁰| if λ6= 0 and λ⁰6= 0, db(n, m, λ),( ˙x, k)

=db( ˙x, k),(n, m, λ)

def=|λ(n+m)−x|˙ 1+|m−n−k|1+|λ| if λ6= 0, db( ˙x, k),( ˙x⁰, k⁰)

=|x˙−x˙⁰|1+|k−k⁰|1.

Proof. — Consider a Cauchy sequence (np, mp, λp)_p∈N in (He^d,d). Ifb p and p⁰ are large enough, then |(mp −n_p)−(m_p⁰ −n_p⁰)| is less than 1, and thus mp−np has to be a constant, that we denote by k. Next, we see that (λ_p)_p∈_Nis a Cauchy sequence of real numbers, and thus converges to some λ in R. If λ 6= 0 then our definition of dbimplies that the sequence (np)_p∈_N is constant after a certain index, and thus converges to someninN^d.Therefore we have (np, mp, λp)→(n, n+k, λ).

Ifλ= 0 then the Cauchy sequence λp(np+mp)

p∈Nhas to converge to some ˙xinR^d.By definition of the extended distance, it is clear that

(np, mp, λp)_p∈_N→( ˙x, k) in Hb^d.

Now, if ˙x6= 0 then there exists some indexj such that ˙xj 6= 0. Because the sequence λ_p(n_j,p+m_j,p)

p∈N tends to ˙x_j and n_j,p+m_j,p is positive (for large enoughp), we must have sgn(λ_p) = sgn( ˙x_j). Therefore, all the of ˙x have the same sign.

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Conversely, let us prove that any point ofR^d+×Z^d (the case ofR^d−×Z^d being similar) is the limit in the sense ofdbof some sequence (np, mp, λp)p∈N. AsQ is dense inR, there exist two families of sequences of positive integers (a_j,p)p∈N and (b_j,p)p∈N such that

∀j∈ {1, . . . , d}, x˙j = lim

p→∞x˙j,p with x˙j,p def= aj,p

bj,p

and lim

p→∞bj,p=∞.

Let us write that

˙

x_p= 2λ_pn_p with λ_p^def=

2

d

Y

j=1

b_j,p −1

and n_p^def=

a_j,p

d

Y

j⁰6=j

b_j,p

16j6d

. As (λ_p)_p∈_Ntends to 0, we have that lim_p→∞db(n_p, n_p+k, λ_p),( ˙x, k)

con-

verges to 0.

Remark 2.6. — It is not difficult to check that the closed bounded subsets ofHb^d (for the distanced) are compact. The details are left to the reader.b

The above proposition prompts us to extend the Fourier transform of an integrable function, to the frequency setHb^d, that will play the same role as (Rⁿ)^? in the case of Rⁿ. With this new point of view, we expect the Fourier transform of any integrable function to be continuous on the whole Hb^d.This is exactly what is stated in the following theorem.

Theorem 2.7. — The Fourier transformfb_H of any integrable function onH^d may be extended continuously to the whole set Hb^d. Still denoting byfb_H (or F_Hf) that extension, the linear map F_H:f 7→fb_H is continuous from the spaceL¹(H^d)to the space C0(Hb^d)of continuous functions onHb^d tending to0at infinity. Moreover, we have for all( ˙x, k)inHb^d0,

F_Hf( ˙x, k) = Z

T^?R^d

Kd( ˙x, k, Y)f(Y, s)dYds with (2.21)

Kd( ˙x, k, Y) =

d

O

j=1

K( ˙xj, kj, Yj) and K( ˙x, k, y, η)^def= 1

2π Z π

−π

e^{i 2|˙}^x|

12(ysinz+ηsgn( ˙x) cosz)+kz dz.

(2.22)

In other words, for any sequence (np, λp)_p∈_NofN^d×(R\ {0}) such that

p→∞lim λp= 0 and lim

p→∞λpnp= x˙ 2, we have

p→∞lim fb_H(np, np+k, λp) = Z

T^?R^d

Kd( ˙x, k, Y)f(Y, s)dYds.

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Granted with the above result, one can propose a natural extension of the Fourier transform to (smooth) functions onH^d independent of the vertical variables.This will come up as a consequence of the following theorem.

Theorem 2.8. — Let us define the following operatorG_HonL¹(T^?R^d):

G_Hg( ˙x, k)^def= Z

T^?R^d

Kd( ˙x, k, Y)g(Y)dY.

Then, for any functionχinS(R)with value1at0and compactly supported Fourier transform, and any functiong inL¹(T^?R^d), we have

(2.23) lim

ε→0F_H(g⊗χ(ε·)) = 2π(G_Hg)µ bH^d₀

in the sense of measures onHb^d, where µ

bH^d₀ is the measure (supported in Hb^d0) defined for all continuous compactly supported functionsθ onHb^d by

(2.24) hµ

bH^d0

, θi= Z

bH^d0

θ( ˙x, k)dµ bH^d0

( ˙x, k)

def= 2^−d X

k∈Z^d

Z

(R−)^d

θ( ˙x, k)d ˙x+ Z

(R+)^d

θ( ˙x, k)d ˙x

·

The above theorem allows to give a meaning of the Fourier transform of a smooth function that does not depend on the vertical variable. The next step would be to study whether our approach allows, as in the Euclidean case, to extend the definition of the Fourier transform to a much larger set of functions, or even to tempered distributions. This requires a fine characterization ofF_H(S(H^d)) the range ofS(H^d) byF_H,which will be the purpose of our companion paper [2].

We end this section with a short description of the structure of the rest of the paper, and of the main ideas of the proofs.

Section 3 is devoted to the proof of the first part of Theorem 2.7. It relies on the fact that the function W(·, Y) is uniformly continuous (for distance d) on bounded sets ofb He^d, and can thus be extended to the clo- sure Hb^d of He^d. Establishing that property requires our using an explicit asymptotic expansion ofW.

Proving Theorem 2.8 is the purpose of Section 4. The main two ingredi- ents are the following ones. First, we show that ifψis an integrable function onRwith integral 1,then we have

ε→0lim 1 εψλ

ε

dwb=µ bH^d₀,

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in the sense of measures onHb^d.That is to say, for any continuous compactly supported functionθ onHb^d,we have

(2.25) lim

ε→0

Z b^H^d

1 εψλ

ε

θ(w)db wb=hµ bH^d0

, θ|bH^d0

i.

Then by a density argument, the proof of Theorem 2.8 reduces to the case whengis in S(T^?R^d).

Section 5 is devoted to computingK. This will be based on the following properties (that we shall first establish):

• K(0, k, Y) =δ0,k for allY inT^?R;

• The symmetry identities:

(2.26) K( ˙x,−k,−Y) =K( ˙x, k, Y), K(−x,˙ −k, Y) = (−1)^kK( ˙x, k, Y) and K(−x, k, Y˙ ) =K( ˙x, k, Y);

• The identity

(2.27) ∆YK( ˙x, k, Y) =−4|x|K( ˙˙ x, k, Y);

• The relation

(2.28) ikK( ˙x, k, Y) = η∂yK( ˙x, k, Y)−y∂ηK( ˙x, k, Y)

sgn( ˙x);

• The convolution property (2.29) K( ˙x, k, Y1+Y2) = X

k⁰∈Z

K( ˙x, k−k⁰, Y1)K( ˙x, k⁰, Y2);

• And finally, the following relation for ˙x > 0 given by the study ofF_H(|Y|²f):

(2.30) |Y|²K+ ˙x∂²_x_˙K+∂x˙K − k² 4 ˙xK= 0.

Let us emphasize that proving first (2.25) is essential to justify rigor- ously (2.22).

Finally, Section 6 is devoted to the proof of an inversion formula involving Operator G_H. Some basic properties of Hermite functions and of Wigner transform of Hermite functions are recalled in Appendix. There, for the reader convenience, we also prove the decay result stated in Lemma 2.3.

3. The uniform continuity of the Fourier transform of an L¹ function

The key to the proof of Theorem 2.7 is a refined study of the behavior of functionsW(·, Y) defined by (2.11) on the setHe^d.Of course, a special

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attention will be given to the neighborhood ofHb^d0. This is the aim of the following proposition.

Proposition 3.1. — LetR0 be a positive real number, and let B(R0)^def=

(n, m, λ)∈He^d, |λ|(|n+m|+d) +|n−m|6R0

×

Y ∈T^?R^d, |Y|6R₀

· The function W(·, Y) restricted to B(R₀) is uniformly continuous with respect tow,b that is for all∀ε >0,there existsαε>0such that

∀(wb_j, Y)∈ B(R₀), d(bwb₁,wb₂)< α_ε=⇒

W(wb₁, Y)− W(wb₂, Y) < ε.

Furthermore, for any( ˙x, k)inHb^d0,we have lim

w→( ˙b x,k)

W(w, Yb ) =K_d( ˙x, k, Y) where the functionKd is defined on Hb^d0×T^?R^d by (3.1) Kd( ˙x, k, Y)^def= X

(`1,`2)∈N^d×N^d

(iη)^`¹

`1! y^`²

`2! F`₁,`₂(k) (sgn ˙x)^`¹|x|˙ ^`^{1 +}²^`² with F_`₁_,`₂(k)^def= X

`⁰₁6`1,`⁰₂6`2

k+`₁−2`⁰₁=`₂−2`⁰₂

(−1)^`²^−`⁰² `1

`⁰₁

`2

`⁰₂

.

Above, sgn ˙x designates the (common) sign of all components of x,˙ and

|x|˙ ^def= (|x˙1|, . . . ,|x˙d|).

Proof. — Let us first perform the change of variablez⁰ =−y+zin (2.11) so as to get

(3.2)

W(w, Yb ) = e^2iλhη,yiW(fw, Yb ) with W(f w, Yb )^def=

Z

R^d

e^2iλhη,z⁰ⁱH_n,λ(2y+z⁰)H_m,λ(z⁰)dz⁰.

Obviously, the uniform continuity ofWreduces to that ofW. Moreover,f as the integral defining Wf is a product of d integrals on R (of modulus bounded by 1), it is enough to study the one dimensional case.

Let us start with the case where both wb1 = (n1, m1, λ1) and wb2 = (n2, m2, λ2) are relatively far away from Hb¹0. As we need only to consider

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the situation where wb₁ and wb₂ are close to one another, one may assume that (n1, m1) = (n2, m2) = (n, m).Then we can write that

W(fwb1, Y)−W(fwb2, Y)

= Z

R

e^2iλ¹^ηz−e^2iλ²^ηz

Hn,λ₁(2y+z)Hm,λ₁(z)dz +

Z

R

e^2iλ²^ηz Hn,λ₁(2y+z)−Hn,λ₂(2y+z)

Hm,λ₁(z)dz +

Z

R

e^2iλ²^ηzH_n,λ₂(2y+z) H_m,λ₁(z)−H_m,λ₂(z) dz.

Clearly, we have (3.3)

Z

R

e^2iλ¹^ηz−e^2iλ²^ηz

Hn,λ₁(2y+z)Hm,λ₁(z)dz

62|λ₁|⁻¹²R₀|λ₂−λ₁| kM H_mk_L2. Next, let us study the continuity of the map λ 7−→ H_n,λ in the case d= 1.One may write

Hn,λ₁(x)−Hn,λ₂(x)

= |λ1|¹⁴ − |λ2|¹⁴

Hn(|λ1|¹²x)

+|λ2|¹⁴ Hn(|λ1|¹²x)−Hn(|λ2|¹²x)

= |λ1|¹⁴ − |λ2|¹⁴

H_n(|λ1|¹²x) +|λ2|¹⁴(|λ1|¹² − |λ2|¹²)x

Z 1 0

H_n⁰ (|λ2|¹² +t(|λ1|¹² − |λ2|¹²))x dt.

If

|λ1|¹² − |λ2|¹²

6 ¹2|λ2|¹²,then the changes of variable x⁰ =|λ1|¹²x and x⁰ = (|λ2|¹² +t(|λ1|¹² − |λ2|¹²))x, respectively, together with the fact that the Hermite functions haveL² norms equal to 1 ensure that

(3.4) kHn,λ₁−Hn,λ₂k_L2 6

|λ1|¹⁴ − |λ2|¹⁴

|λ1|¹⁴ + 4

|λ1|¹² − |λ2|¹²

|λ2|¹² kM H_n⁰k_L2. Using (A.4), we get that

M H_n⁰ = 1 2

pn(n−1)Hn−2−Hn−p

(n+ 1)(n+ 2)Hn+2

. As the family of Hermite functions is an orthonormal basis ofL², one can write that

4kM H_n⁰k²_L2 =n(n−1) + 1 + (n+ 1)(n+ 2) = 2n²+ 2n+ 3.

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Hence, one can conclude that if

|λ1| − |λ2|| 6 ¹₂|λ2|¹² |λ1|¹² +|λ2|¹² and (|λ1|+|λ2|)|n+m|+|λ1|+|λ2|+|Y|6R0 then

(3.5)

fW(wb₁, Y)−W(fwb₂, Y)

6C(R₀)|λ₁−λ₂| 1

|λ1|+ 1 λ²₁

· That estimate fails if the above condition on

|λ1| − |λ2|

is not satisfied.

To overcome that difficulty, we need the following lemma.

Lemma 3.2. — The series X

`1,`2

(sgnλ)^`¹|λ|^`^{1 +}²^`² (2iη)^`¹(2y)^`²

`1!`2! M^`¹H_m|∂^`²H_n

L²(R^d)

converges normally towardsWfonB(R0).

Proof. — Again, as Hermite functions in dimension dare tensor products of one-dimensional Hermite functions, it is enough to prove the above lemma in dimension 1. Now, using the expansion of the exponential function and Lebesgue theorem, we get that for any fixed (w, Yb ) inHe¹×T^?R,

W(fw, Yb ) =

∞

X

`1=0

1

`1!(2iλη)^`¹ Z

R

Hn,λ(2y+z)z^`¹Hm,λ(z)dz

=

∞

X

`1=0

(sgnλ)^`¹(2iη)^`¹

`1! |λ|^`²¹ Z

R

Hn,λ(2y+z)(M^`¹Hm)λ(z)dz.

(3.6)

Let us prove that the series converges for the supremum norm on B(R0).

Clearly, (A.4) implies that for all integers`>1 andminN, k(√

2M)^`Hmk_L2(R)6√ mk(√

2M)^`−1H_m−1k_L2(R)

+√

m+ 1k(√

2M)^`−1Hm+1k_L2(R), which, by an obvious induction yields for all (`1, m) inN²,

(3.7) kM^`¹Hmk_L2(R)6(2m+ 2`₁)^`²¹.

Hence the generic term of the series of (3.6) can be bounded by:

W`₁(w)b ^def= (2√ 2R0)^`¹

`₁! |λ|^`²¹(m+`1)^`²¹.

Let us observe that, because|λ|mand|λ|are less thanR0, we have W`₁+1(w)b

W_`₁(w)b =2√ 2R0

`₁+ 1

p|λ|(m+`1+ 1)

1 + 1

m+`₁ ^`₂¹

62√ 2e R0

`₁+ 1

pR0 1 +p

`1+ 1 .

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This implies that the series converges with respect to the supremum norm onB(R0).

Next, for fixed`₁,we want to expand

|λ|^`²¹ Z

R

Hn,λ(2y+z)(M^`¹Hm)λ(z)dz

as a series with respect to the variable y. To this end, we just have to expand the real analytic Hermite functions as follows:

Hn,λ(z+ 2y) =

∞

X

`2=0

(2y)^`²

`2! |λ|^`²²(H_n^(`²⁾)λ(z).

Then we have to study (for fixed `₁) the convergence of the series with general term,

W`₁,`₂(w, Yb )^def= (2y)^`²

`₂! |λ|^`²² H_n^(`²⁾|M^`¹Hm

L². Using again (A.4), we see that

kH_n^(`²⁾k_L2(R)6(2n+ 2`₂)^`²². Hence, arguing as above, we get for any (w, Yb ) inB(R0),

W_`₁_,`₂(w, Yb )62^`²¹(m+`₁)^`²¹fW_`₂(w, Yb )

with fW`₂(w, Yb )^def= (2√ 2R0)^`²

`₂! |λ|^`²²(n+`2)^`²², and it is now easy to complete the proof of the lemma.

Reverting to the proof of the continuity ofWfin the neighborhood ofHb^d0, the problem now consists in investigating the behavior of the function

H`₁,`₂ : (

He¹ −→R

wb= (n, m, λ)7−→ |λ|^`^{1 +}²^`² M^`¹Hm|Hn^(`²⁾

L²(R)

whenλtends to 0 andλ(n+m)→x˙ for fixedk^def=m−n.

From Relations (A.3), we infer that

H`₁,`₂(w) = 2b ^−(`¹^+`²⁾|λ|^`^{1 +}²^`² (A+C)^`¹Hm|(A−C)^`²Hn

L²(R). The explicit computation of (A±C)^`is doable but tedious and fortunately turns out to be useless whenλtends to 0. Indeed, we have the following lemma: